Most Probable Paths for Anisotropic Brownian Motions on Manifolds

Abstract

Brownian motion on manifolds with non-trivial diffusion coefficient can be constructed by stochastic development of Euclidean Brownian motions using the fiber bundle of linear frames. We provide a comprehensive study of paths for such processes that are most probable in the sense of Onsager–Machlup, however with path probability measured on the driving Euclidean processes. We obtain both a full characterization of the resulting family of most probable paths, reduced equation systems for the path dynamics where the effect of curvature is directly identifiable, and explicit equations in special cases, including constant curvature surfaces where the coupling between curvature and covariance can be explicitly identified in the dynamics. We show how the resulting systems can be integrated numerically and use this to provide examples of most probable paths on different geometries and new algorithms for estimation of mean and infinitesimal covariance.

Appendices

Definition of Sub-Riemannian Geometry

We give a quick introduction to sub-Riemannian geometry and refer to [10] for details. A sub-Riemannian manifold is a triple $(M, E,\rho )$ where M is a connected manifold, E is a subbundle of the tangent bundle TM and $\rho = \langle \cdot , \cdot \rangle _\rho$ is a metric tensor defined only on E. This tensor defines a vector bundle morphism $\sharp ^\rho : T^*M \rightarrow E \subseteq TM$ given by

$$\begin{aligned} \alpha (v) = \langle \sharp ^\rho \alpha , v \rangle _\rho . \end{aligned}$$

Consequently, we obtain a positive semi-definite symmetric tensor $\rho ^* = \langle \cdot , \cdot \rangle _{\rho ^*}$ on $T^*M$ defined by

$$\begin{aligned} \langle \alpha , \beta \rangle _{\rho ^*} = \langle \sharp ^\rho \alpha , \sharp ^\rho \beta \rangle _\rho . \end{aligned}$$

This tensor degenerates along the subbundles ${{\,\mathrm{Ann}\,}}(E) \subseteq T^*M$ of covectors vanishing on E. It follows that a sub-Riemannian manifold can equivalently be defined as a connected manifold with positive, semi-definite cometric $\rho ^*$ that degenerates along a subbundle.

An absolutely continuous curve $\gamma :[0,T] \rightarrow M$ is called horizontal if ${{\dot{\gamma }}}(t) \in E_{\gamma (t)}$ for almost every t. For such a curve, we define its length to be

$$\begin{aligned} L^\rho (\gamma ) = \int _0^T | {{\dot{\gamma }}}|_\rho (t)\, {d}t. \end{aligned}$$

This length is invariant under reparametrization, so we can restrict our considerations to the case $T =1$.

For any $x, y \in M$, we define

$$\begin{aligned} d_\rho (x,y) = \inf \left\{ L^\rho (\gamma ) \, : \, \begin{array}{c} \gamma :[0,1] \rightarrow M \text { horizontal} \\ \gamma (0) = x, \gamma (1) = y \end{array} \right\} . \end{aligned}$$

We notice that if there are no horizontal curves connecting the two points, then $d_\rho (x,y) =\infty$. Let $x \in M$ be a given point. We define ${\mathcal {C}}(x)$ as the space of all horizontal curves defined on [0, 1], with $L^2$-derivative, that start in x. This collection has a natural structure of a Hilbert manifold, see [10, Chapter 5.1] for details. Define a mapping

$$\begin{aligned} \varPi : {\mathcal {C}}(x) \rightarrow M, \qquad \gamma \mapsto \gamma (1). \end{aligned}$$

Define ${\mathcal {C}}(x,y) = \varPi ^{-1}(y)$. A point $\gamma \in {\mathcal {C}}(x,y)$ is called regular if $\varPi _{*,\gamma }:T_\gamma {\mathcal {C}}(x) \rightarrow T_y M$ is surjective. Otherwise, $\gamma$ is called singular or abnormal curves.

Assume that ${\mathcal {C}}(x,y)$ non-empty. Define $F: {\mathcal {C}}(x,y) \rightarrow {\mathbb {R}}$ by $\gamma \mapsto L^\rho (\gamma )$. We look at minimal elements in ${\mathcal {C}}(x,y)$ with respect to F. If $\gamma$ is a regular curve, then ${\mathcal {C}}(x,y)$ locally has the structure of a Hilbert manifold around $\gamma$ by the inverse function theorem. Hence, any regular minimal element must be a critical, i.e., we must have $F_{*,\gamma } = 0$. Such curves are called normal geodesics, and will always be locally length minimizing. It can be shown that all such curves, up to reparametrization, be found as a projecting of a solution of a Hamiltonian system. The Hamiltonian is given by

$$\begin{aligned} P(\alpha ) = \frac{1}{2} \langle \alpha , \alpha \rangle _{\rho ^*}, \qquad \alpha \in T^*M. \end{aligned}$$

In conclusion, length minimizers are either normal geodesics or abnormal curves. These classes of curves are not necessarily disjoint.

We say that E is bracket-generating if for every point $x \in M$,

$$\begin{aligned} {{\,\mathrm{span}\,}}\{ X_i, [X_i,X_j], [X_i, [X_j, X_k]], \dots , \} |_x \in T_xM, \qquad X_i \in \varGamma (E), \end{aligned}$$

that is, if sections of E generate the entire tangent bundle TM. If this condition holds, then any pair of points can be connected by a horizontal curve. The value of $d_\rho$ is always finite, and furthermore, it induces the same topology as the manifold topology.

Remark 7

Let L be a second order operator on M without constant term, such that for any pair of smooth functions $f,g \in C^\infty (M)$,

$$\begin{aligned} L(fg) - fLg - gL f = \langle df, dg \rangle _{\rho ^*}. \end{aligned}$$

In other words, locally, L can always be written as $L = \sum _{j=1}^{{{\,\mathrm{rank}\,}}E} V_i^2 + V_0$, where $V_1, \dots , V_{{{\,\mathrm{rank}\,}}E}$ is a local orthonormal basis of $(E,\rho )$. If E is bracket generating, then L is hypoelliptic [7] and its heat semigroup $p_t(x;y)$ has a strictly positive density [19].

Sub-Riemannian Normal Geodesics on $({{\,\mathrm{Sym}\,}}^+ TM, E, \rho )$

By the discussion in Sect. 4.2, it follows that we can write a normal geodesic as $\varSigma (t) = f(t)^{-1} S^2 f(t)$ where f(t) is a normal geodesic in $({{\,\mathrm{O}\,}}(TM), {\mathcal {H}}, \rho _S)$. We do the computations here.

Recall the definition of the vector fields $H_a$, $a \in {\mathbb {R}}^n$ and $\xi _A$, $A \in {{\,\mathrm{\mathfrak {so}}\,}}(n)$ in Sect. 3. We introduce corresponding Hamiltonian functions

$$\begin{aligned} P_a(\alpha ) = \alpha (H_a), \qquad Q_A(\alpha ) = \alpha (\xi _A), \qquad \alpha \in T^* {{\,\mathrm{O}\,}}(TM). \end{aligned}$$

Our formulas in (3.3), then give corresponding relations in terms of Poisson brackets

$$\begin{aligned} \{ P_a, P_b\} = Q_{{\underline{R}}(a,b)}, \qquad \{ Q_A, P_a\} = - P_{Aa}, \qquad \{ Q_A, Q_B\} = - Q_{[A,B]}. \end{aligned}$$

Since $H_{Se_1}, \dots , H_{Se_d}$ is a global orthonormal basis, we have that the sub-Riemannian Hamiltonian is given by

$$\begin{aligned} P = \frac{1}{2} \sum _{j=1}^d P_{Se_j}^2. \end{aligned}$$

Let $\lambda (t) = e^{t\mathbf {P}}(\lambda _0)$ be a solution in $T^* FM$ along f(t) in FM and define curves v(t) in ${\mathbb {R}}^d$ and A(t) in ${{\,\mathrm{\mathfrak {so}}\,}}(d)$ by

$$\begin{aligned} P_a(t) = P_a(\lambda (t)) = \langle S^{-2} v(t), a \rangle , \qquad Q_B(t) = Q_B(\lambda (t)) = - \langle A(t), B \rangle . \end{aligned}$$

Then along a solution, we have

$$\begin{aligned} \langle S^{-2} \dot{v}, a \rangle&= \dot{P}_a = \{ P_a, P\} = \sum _{j=1}^d P_{Se_j} Q_{{\underline{R}}(a, Se_j)}\\&= - \sum _{j=1}^d \langle S^{-2} v, Se_j \rangle \langle A, {\underline{R}}(a, Se_j) \rangle \\&= \sum _{j=1}^d \langle S^{-1} v, e_j \rangle \langle {\underline{R}}(A)S e_j, a\rangle = \langle {\underline{R}}(A) v, a\rangle \\ - \langle \dot{A}(t) , B \rangle&= - \dot{Q}_B \\&= - \{ Q_B, P \} = \sum _{j=1}^d P_{Se_j} P_{BSe_j} \\&= \sum _{j=1}^d \langle S^{-2}v, Se_j \rangle \langle BS e_j ,S^{-2} v \rangle \\&= \langle B v, S^{-2}v \rangle \\&= \langle v \wedge S^{-2} v, B \rangle . \end{aligned}$$

In summary, $\dot{v} = S^2 {\underline{R}}(A) v$ and $\dot{A} = v \wedge S^{-2} v$.

Let $\beta \in \varGamma (T^* {{\,\mathrm{O}\,}}(TM))$ be a one-form on ${{\,\mathrm{O}\,}}(TM)$ and define the corresponding vertical lift ${{\,\mathrm{vl}\,}}\beta \in \varGamma (T(T^* {{\,\mathrm{O}\,}}(TM)))$ by

$$\begin{aligned} {{\,\mathrm{vl}\,}}\beta |_\alpha = \frac{{d}}{{d}t} (\alpha + t\beta _f) |_{t=0}, \qquad \alpha \in T_f^* {{\,\mathrm{O}\,}}(TM). \end{aligned}$$

Let $\vartheta$ be the Liouville one form $\vartheta |_\alpha = \pi ^* \alpha$ with canonical symplectic form $\sigma = - d\vartheta$. Observe that $\sigma ({{\,\mathrm{vl}\,}}\beta , \, \cdot \, ) = - (\pi ^* \beta )(\, \cdot \,)$. Then

$$\begin{aligned} dP({{\,\mathrm{vl}\,}}\beta )&= \sum _{j=1}^d P_{S e_j} \beta (H_{S e_j} ) = \sigma ( \mathbf {P}, {{\,\mathrm{vl}\,}}\beta ) = \beta (\pi _* \mathbf {P}). \end{aligned}$$

If we consider this along the curve, we have

$$\begin{aligned} \sum _{j=1}^d P_{S e_j} \beta (H_{S e_j} ) |_{\lambda (t)}&= \sum _{j=1}^d \langle S^{-1}v(t), e_j \beta (H_{S e_j} ) |_{f(t)} = \beta (H_{v(t)} ) |_{f(t)} \\&= \beta (\pi _* \mathbf {P}) |_{\lambda (t)} = \beta (\dot{f}(t)) \end{aligned}$$

It follows that

$$\begin{aligned}&\dot{f}(t) = H_{v(t)}, \\&\dot{v}(t) = S^2 {\underline{R}}(f(t))(A(t)) v(t), \\&\dot{A}(t) =v(t) \wedge S^{-2} v(t). \end{aligned}$$

We see that these are exactly the equations of found in the proof of Theorem 1 without the condition $A(T) = 0$.